Symbolic & Numeric Calculus

Solve a SIAM Challenge Problem

The integral depends on the parameter α. Find the value of that lies between and and maximizes the integral. The given integral can be regarded as a Mellin convolution of two functions.

In[1]:=
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f[x_] := x (2 - x)^\[Alpha] UnitBox[(x - 1)/2]
In[2]:=
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g[x_] := Sin[x]

Compute the Mellin convolution of f[x] and g[x].

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(mc = MellinConvolve[f[x], g[x], x, \[Alpha]]) // TraditionalForm
Out[3]//TraditionalForm=

Compare with the result given by Integrate.

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Integrate[(2 - x)^\[Alpha] Sin[\[Alpha]/x], {x, 0, 2}, Assumptions -> \[Alpha] > 0] // TraditionalForm
Out[4]//TraditionalForm=

Plot the integral as a function of .

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Plot[mc // Evaluate, {\[Alpha], 0, 4.99}, PlotStyle -> Red]
Out[5]=

Compute the argument that maximizes the integral in using FindArgMax.

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N[FindArgMax[mc, {\[Alpha], 1}, WorkingPrecision -> 100][[1]], 20]
Out[6]=

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